Rangers recorded an emphatic victory, but were only able to breathe easily after their fourth goal went in.
Lee McCulloch put Rangers ahead with a penalty before Philipp Zufle was red-carded for a challenge on Kevin Kyle.
Andrew Little volleyed in Rangers' second but Kevin Turner pulled one back before half-time.
Lee Wallace scored early in the second half countered by a Paul Quinn penalty. But Kyle, Kal Naismith and another McCulloch penalty secured the win.
Shire were competitive throughout, only fading when Rangers' superior fitness and the extra man began to tell.
Ally McCoist's side were impressive in spells, with Barrie McKay on particularly good form.
The visitors put together an early enterprising move involving Wallace, McKay and Ian Black down the left, but Wallace's cross was fired wide by Dean Shiels.
But Shiels produced a terrific, dancing run across the edge of the Shire box before being cut down by Sean Kelly just inside to win his side a penalty, which was successfully despatched by McCulloch.
That breakthrough came on 11 minutes and, shortly afterwards, Rangers' advantage should have been doubled.
Kyle flicked on for Shiels who in turn slipped it to McKay. The winger's lob left the keeper beaten but Little somehow guided the ball wide.
The game hit something of a lull but was reignited as Zufle was dismissed for a reckless challenge on Kyle.
Black, looking sharper than he has previously this season, then brought out a good save from Hay, but the keeper could do nothing about Rangers' second goal on 38 minutes.
Little collected the ball in the right channel with one touch before hammering it goalwards and with the aid of a deflection off Ricky Miller it looped over Hay.
But almost immediately the home side pulled a goal back. Nathan Shepherd whipped in a terrific cross from the left, which was met emphatically by the head of Turner.
Any half-time thoughts John Coughlin's side may have had about taking something from the match appeared to be quickly extinguished.
Four minutes after the restart, Wallace waltzed down the left, past a couple of challenges before striking the ball right-footed past Hay from the edge of the area.
The Shire were not finished yet, though. Shepherd was clipped inside the box by McCulloch and Quinn, as he did at Ibrox, beat Neil Alexander from the penalty spot.
But no sooner had the home side restored a glimmer of hope, than Rangers scored their fourth goal of the afternoon.
Shiels drifted in from the left, played the ball into the path of Little who failed to connect properly with his shot, but Kyle was on hand to tuck it past Hay, who had to pick the ball out the back of the net again on 71 minutes.
This time it was McKay who raced down the left before providing for substitute Naismith, who placed the ball beyond the keeper's reach and in at the far post.
McKay was the best player on the park and he won Rangers' second penalty with a piece of trickery that led Steven Jackson to handle. McCulloch again did the business from 12 yards.
Hay, Jackson, Miller, Sean Kelly, Shepherd, Greenhill (Hunter 67), Zufle, McKernon, Maxwell (Herd 65), Turner, Quinn (Begg 73). Subs Not Used: Gordon, Hume.
Turner 41, Quinn 58 pen.
Alexander, Argyriou, McCulloch, Hegarty, Wallace, Little (Aird 78), Black, Macleod (Crawford 83), McKay, Shiels (Naismith 68), Kyle. Subs Not Used: Scott Gallagher, Hutton.
McCulloch 12 pen, Little 39, Wallace 49, Kyle 63, Naismith 71, McCulloch 82 pen.